"Mathematics in search of mathematicians" was the headline for an article by Stéphane Foucart in the December 4 2009 Le Monde. Foucart was reporting on the two-day "Math to Come" colloquium, which had been held in Paris on the first and second of that month. The agenda was a discussion of the state of mathematics, in France but also in general, and the design of appropriate plans for the future. "At the center of the discussions was a surprising paradox: while mathematics are more and more necessary for the functioning of the world, they are, at least in rich countries, less and less popular with students." Even in France, where the mathematical tradition is "among the most brilliant," students are dwindling: some universities have lost a third of their graduate math enrollment over the last six years. The question is, why? Some blame it on the recent financial crisis, but in fact "financial mathematics is responsible for up to a quarter of the French Masters-level students today." Foucart also mentions problems with the structure of public support for research: "These questions are all the more crucial for math since math alone has the magic to suddenly transform what can appear to be a pure mental game into the indispensable key for solving new, applied problems." [My translations; it is unfortunate that neither this article (available online), nor the Math to Come colloquium, seems to try and pinpoint what in modern western culture steers potential scientists away from useful and rewarding careers. -TP]

Basic instinct: math

Discover's November 2009 issue featured a report by Carl Zimmer with the title "Humanity's Other Basic Instinct: Math" and the subtitle "New research suggests that math has evolved its way right into our neurons --and monkeys' too." Zimmer sketches some of the known history of mathematics, and then writes: "Despite the late appearance of higher mathematics, there is growing evidence that numbers are not really a recent invention -- not even remotely. ... our species seems to have an innate skill for math, a skill that may have been shared by our ancestors going back least 30 million years."

Some of the evidence comes from the study of newborn humans. Zimmer links to a study published last June (PNAS 106 10382-10385, abstract) by Véronique Izard and collaborators. "She and her colleagues played cooing sounds to babies, with varying numbers of sounds in each trial. The babies were then shown a set of shapes on a computer screen ... Newborns consistently looked longer at the screen when the number of shapes matched the number of sounds they had just heard. ... Izard's study suggests that newborns already have a basic understanding of numbers. Moreover, their concept of numbers is abstract; they can transfer it across the senses from sounds to pictures."

Other evidence comes from tests on adults: A team at Duke (Jessica Cantlon, Michael Platt, Elizabeth Brannon) published "Beyond the number domain" last January (abstract) in Trends in Cognitive Sciences. "Cantlon ... ran an experiment in which adult subjects see a set of dots on a computer screen for about half a second, followed by a second set. After a pause, the participants see two sets of dots side by side. They then have a little more than a second to pick the set that is the sum of the previous two pictures. People do fairly well on these tests, which summons up a weird feeling in them: They know they are right, but they don't know how they got the answer." The explanation: "the brain automatically processes numbers."

Zimmer argues that since infants are born with mathematical intuition, our evolutionary ancestors must have had it too. "Scientists have found that many primates, including rhesus monkeys, can solve some of the same mathematical problems we can. Since monkeys and humans diverged 30 million years ago, mathematical intuition presumably is at least that old." Cantlon and her team "were able to teach monkeys to do addition by intuition the same way people do. The animals' intuition is about as good as ours, and it follows the same rules. ... And when monkeys use their mathematical intuition, they rely on the same region of the brain around the intraparietal sulcus that we do."

Cognitive science applied to teaching math

The interparietal sulcus also shows up in a New York Times article (December 20, 2009): "Studying Young Minds, and How to Teach Them." There Benedict Carey explains how "findings, mostly from a branch of research called cognitive neuroscience, are helping to clarify when young brains are best able to grasp fundamental concepts." He quotes Kurt Fisher, director of the Mind, Brain and Education program at Harvard: "... for the first time we are seeing the fields of brain science and education work together." In mathematics, this means starting early to develop children's innate apprehension of number (see previous item) into the precise tool they need to succeed in kindergarten and beyond. "By preschool, the brain can handle larger numbers and is struggling to link three crucial concepts: physical quantities (seven marbles, seven inches) with abstract digit symbols ("7"), with the corresponding number words ("seven" )." To show us how this works, Carey takes us into a classroom in Buffalo where Mrs. Pat Andzel is leading her preschoolers, over and over in different and often entertaining contexts, through the algorithm of counting: the name of the number of things in some set is the last word you pronounce when you count them. "Many of these kids don't understand that yet," she says.

Back to Antikythera

Tony Freeth, the spokesman for the team investigating the Antikythera mechanism, brings us up to date on the project with "Decoding an Ancient Computer" in the December 2009 Scientific American. He presents to the general public and with new pictorial detail the research described in two Nature reports: 444, 587-591, November 30, 2006; 454, 614-617, July 31, 2008, the second of which was picked up here. Until this team went to work, the last word on the mechanism had mostly been Derek da Solla Price's 1974 article in the Transactions of the American Philosophical Society, previewed in Scientific American in June, 1959. Our Feature Columns (April and May, 2000) were based on Price's analysis. The technology available to Freeth and his collaborators, including computed X-ray tomography (like a high-resolution CAT-scan), has allowed a much better understanding of the structure of the mechanism and has proved that Price, while correctly assessing the intellectual scale of the Greek's achievement, was dead wrong on many of the details. In particular the differential gear assembly, which Bill Casselman so lovingly animated for the May column, turns out to have been a figment of Price's imagination: his interpolated idler wheel is an especially creative bit of wishful thinking. The actual assembly is in fact a differential: it contains an epicyclic gear (one mounted off-center on another); as Freeth remarks: "Epicyclic gears extend the range of formulas gears can calculate beyond multiplications of fractions to additions and subtractions." But this differential incorporates an equally ingenious wrinkle, which Freeth terms "a conception of pure genius:" two identical gears connected by a slot-and-pin assembly which makes one move alternately slightly faster, and slightly slower, than the other; in the Antikythera mechanism this portrays the monthly variation in the Moon's angular velocity, which the Greeks had observed. Today we explain it by Kepler's third law.